Local existence of K-sets, projective tensor products, and Arens regularity for A(E-1+center dot center dot center dot+E-n)

摘要

Theorem. If X-1,...,X-n are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets Y-1 subset of or equal to X-1,...,Y-n subset of or equal to X-n such that (i) Y-j is either a Kronecker set or (ii) for some pj greater than or equal to 2, Y-j is a translate of a K-pj-set all of whose elements have order pj, and (iii) A(Y-1 +...+ Y-n) is isomorphic to the projective tensor product C(Y-1) (⊗) over cap...(⊗) over capC(Y-n).This extends what was previously known for groups such as T or for the case n = 2 to the general locally compact abelian group. Old results concerning the local existence of Kronecker and K-p-sets are improved.